Answer
Domain = $\{(x,y)|9-x^{2}-9y^{2} \gt 0\}$
(see image)
Work Step by Step
Because of the restriction for logarithmic functions, f is defined when
$9-x^{2}-9y^{2} \gt 0$
The $ \mathbb{R}^{2}$ plane is divided into two regions by the ellipse
$x^{2}+9y^{2}=9.$
$\displaystyle \frac{x^{2}}{3^{2}}+\frac{y^{2}}{1^{2}}=1$
The ellipse itself is not included (inequality is $ \gt $) - graph with dashed line.
The point (0,0) satisfies the inequality, so the domain of f is the region containing (0,0), inside the ellipse.
Domain = $\{(x,y)|9-x^{2}-9y^{2} \gt 0\}$
(see image)
